And almost everyone also accepts that this isn't enough. After all, a paranoid guy believes everyone is out to get him. If two identical twins
had a genetic disease that made them paranoid, but someone really was out to get one of them, it seems crazy to suggest one knows he's being spied on but one only believes it.

Whatever warrant is, the important thing is that it's this mysterious property that's transmitted by known implication in a closed system. In other words, when I say, as above, that Blode "can know it's Monday today", I'm really saying "Blode is warranted to believe it's Monday today, and it is."

Closure seems so trivially true that it hardly seems worth stating it. We use the principle thousands of times every day, and it works. But closure also has a role to play for the evil monster of philosophy; scepticism.

A common formulation of the sceptical problem goes something like this*:

A full discussion of scepticism is outside the scope of this piece. However, I hope it's clear that closure is playing an important part of setting up this argument. Philosophers attack the above argument from all angles, and some accept its conclusion. But because of its intuitive rightness, premise 4 -- that closure holds -- is rarely their target.

Robert Nozick's account of warrant contains tracking conditions for knowledge. The relevant one here is that if P were false, I wouldn't believe P. This condition is not closed under known implication: Take premise 3: "If I'm typing a writeup than I'm not in a Scep."

If I weren't typing a writeup, I wouldn't believe I was. But if I was in a Scep (e.g the Matrix), being forced to believe I was typing a writeup, then I would believe I was. If you accept Nozick's account of knowledge, closure is false in these sorts of cases.

For Nozick, this is the solution to the sceptical problem. For most other people, it's just proof that Nozick's theory of knowledge is wrong.

Contextualists such as Keith DeRose and David Lewis have a halfway house solution that claims closure applies in given contexts but not across contexts. There's no way of properly explaining this on its own; it belongs in a full discussion of Contextualism.